The proto-neutron star produced in a supernova emits powerful fluxes of neutrinos and antineutrinos of all three flavors for approximately 10 seconds. Immediately outside the proto-neutron star, the number density of neutrinos is so high that forward scattering among themselves must be taken into account in order to treat their flavor evolution on their way out of the supernova. This gives rise to the novel phenomena of collective neutrino flavor oscillations that are sensitive to the yet-unknown hierarchy of neutrino masses. Results from numerical simulations of collective neutrino oscillations in realistic supernova environments will be presented. The underlying physics will be illustrated with simple models. The potential of using supernova neutrino signals to determine the neutrino mass hierarchy will be discussed.